Q8:2D28 - GroupNames

G = Q₈⋊₂D₂₈ order 448 = 2⁶·7

1^st semidirect product of Q₈ and D₂₈ acting via D₂₈/D₁₄=C₂

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Q₈⋊₂D₂₈, D₁₄⋊₇SD₁₆, Dic₁₄⋊₄D₄, (C₇×Q₈)⋊₁D₄, C₄.₉₂(D₄×D₇), C₄.₅(C₂×D₂₈), D₁₄⋊C₈⋊₁₃C₂, C₄⋊C₄.₂₃D₁₄, C₇⋊₂(Q₈⋊D₄), Q₈⋊C₄⋊₁₁D₇, C₂₈.₁₂₀(C₂×D₄), C₄⋊D₂₈.₁C₂, (C₂×C₈).₁₂₃D₁₄, C₂.₁₇(D₇×SD₁₆), C₁₄.₂₃C₂²≀C₂, C₁₄.Q₁₆⋊₁₁C₂, (C₂×Q₈).₁₀₇D₁₄, (C₂×Dic₇).₃₀D₄, C₁₄.₃₁(C₂×SD₁₆), (C₂²×D₇).₇₇D₄, C₂².₁₉₆(D₄×D₇), (C₂×C₂₈).₂₄₆C₂³, (C₂×C₅₆).₁₃₄C₂², (C₂×D₂₈).₆₂C₂², (Q₈×C₁₄).₂₉C₂², C₂.₂₆(C₂²⋊D₂₈), C₂.₁₄(Q₁₆⋊D₇), C₁₄.₆₀(C₈.C₂²), (C₂×Dic₁₄).₇₀C₂², (C₂×Q₈×D₇)⋊₁C₂, (C₂×Q₈⋊D₇)⋊₂C₂, (C₂×C₅₆⋊C₂)⋊₁₇C₂, (C₂×C₇⋊C₈).₃₈C₂², (C₂×C₄×D₇).₂₀C₂², (C₇×Q₈⋊C₄)⋊₁₁C₂, (C₂×C₁₄).₂₅₉(C₂×D₄), (C₇×C₄⋊C₄).₄₇C₂², (C₂×C₄).₃₅₃(C₂²×D₇), SmallGroup(448,340)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₂₈ — Q₈⋊₂D₂₈

Chief series

C₁ — C₇ — C₁₄ — C₂×C₁₄ — C₂×C₂₈ — C₂×C₄×D₇ — C₂×Q₈×D₇ — Q₈⋊₂D₂₈

Lower central

C₇ — C₁₄ — C₂×C₂₈ — Q₈⋊₂D₂₈

Upper central

C₁ — C₂² — C₂×C₄ — Q₈⋊C₄

Generators and relations for Q₈⋊₂D₂₈
G = < a,b,c,d | a⁴=c²⁸=d²=1, b²=a², bab^-1=cac^-1=dad=a^-1, cbc^-1=dbd=ab, dcd=c^-1 >

Subgroups: 980 in 158 conjugacy classes, 45 normal (37 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₇, C₈, C₂×C₄, C₂×C₄, D₄, Q₈, Q₈, C₂³, D₇, C₁₄, C₂²⋊C₄, C₄⋊C₄, C₂×C₈, C₂×C₈, SD₁₆, C₂²×C₄, C₂×D₄, C₂×Q₈, C₂×Q₈, Dic₇, C₂₈, C₂₈, D₁₄, D₁₄, C₂×C₁₄, C₂²⋊C₈, Q₈⋊C₄, Q₈⋊C₄, C₄⋊D₄, C₂×SD₁₆, C₂²×Q₈, C₇⋊C₈, C₅₆, Dic₁₄, Dic₁₄, C₄×D₇, D₂₈, C₂×Dic₇, C₂×Dic₇, C₂×C₂₈, C₂×C₂₈, C₇×Q₈, C₇×Q₈, C₂²×D₇, C₂²×D₇, Q₈⋊D₄, C₅₆⋊C₂, C₂×C₇⋊C₈, D₁₄⋊C₄, Q₈⋊D₇, C₇×C₄⋊C₄, C₂×C₅₆, C₂×Dic₁₄, C₂×Dic₁₄, C₂×C₄×D₇, C₂×C₄×D₇, C₂×D₂₈, C₂×D₂₈, Q₈×D₇, Q₈×C₁₄, C₁₄.Q₁₆, D₁₄⋊C₈, C₇×Q₈⋊C₄, C₄⋊D₂₈, C₂×C₅₆⋊C₂, C₂×Q₈⋊D₇, C₂×Q₈×D₇, Q₈⋊₂D₂₈
Quotients: C₁, C₂, C₂², D₄, C₂³, D₇, SD₁₆, C₂×D₄, D₁₄, C₂²≀C₂, C₂×SD₁₆, C₈.C₂², D₂₈, C₂²×D₇, Q₈⋊D₄, C₂×D₂₈, D₄×D₇, C₂²⋊D₂₈, D₇×SD₁₆, Q₁₆⋊D₇, Q₈⋊₂D₂₈

Smallest permutation representation of Q₈⋊₂D₂₈
►On 224 points

Generators in S₂₂₄

(1 161 64 122)(2 123 65 162)(3 163 66 124)(4 125 67 164)(5 165 68 126)(6 127 69 166)(7 167 70 128)(8 129 71 168)(9 141 72 130)(10 131 73 142)(11 143 74 132)(12 133 75 144)(13 145 76 134)(14 135 77 146)(15 147 78 136)(16 137 79 148)(17 149 80 138)(18 139 81 150)(19 151 82 140)(20 113 83 152)(21 153 84 114)(22 115 57 154)(23 155 58 116)(24 117 59 156)(25 157 60 118)(26 119 61 158)(27 159 62 120)(28 121 63 160)(29 190 95 203)(30 204 96 191)(31 192 97 205)(32 206 98 193)(33 194 99 207)(34 208 100 195)(35 196 101 209)(36 210 102 169)(37 170 103 211)(38 212 104 171)(39 172 105 213)(40 214 106 173)(41 174 107 215)(42 216 108 175)(43 176 109 217)(44 218 110 177)(45 178 111 219)(46 220 112 179)(47 180 85 221)(48 222 86 181)(49 182 87 223)(50 224 88 183)(51 184 89 197)(52 198 90 185)(53 186 91 199)(54 200 92 187)(55 188 93 201)(56 202 94 189)

(1 169 64 210)(2 103 65 37)(3 171 66 212)(4 105 67 39)(5 173 68 214)(6 107 69 41)(7 175 70 216)(8 109 71 43)(9 177 72 218)(10 111 73 45)(11 179 74 220)(12 85 75 47)(13 181 76 222)(14 87 77 49)(15 183 78 224)(16 89 79 51)(17 185 80 198)(18 91 81 53)(19 187 82 200)(20 93 83 55)(21 189 84 202)(22 95 57 29)(23 191 58 204)(24 97 59 31)(25 193 60 206)(26 99 61 33)(27 195 62 208)(28 101 63 35)(30 155 96 116)(32 157 98 118)(34 159 100 120)(36 161 102 122)(38 163 104 124)(40 165 106 126)(42 167 108 128)(44 141 110 130)(46 143 112 132)(48 145 86 134)(50 147 88 136)(52 149 90 138)(54 151 92 140)(56 153 94 114)(113 188 152 201)(115 190 154 203)(117 192 156 205)(119 194 158 207)(121 196 160 209)(123 170 162 211)(125 172 164 213)(127 174 166 215)(129 176 168 217)(131 178 142 219)(133 180 144 221)(135 182 146 223)(137 184 148 197)(139 186 150 199)

(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28)(29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84)(85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112)(113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140)(141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168)(169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196)(197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224)

(1 63)(2 62)(3 61)(4 60)(5 59)(6 58)(7 57)(8 84)(9 83)(10 82)(11 81)(12 80)(13 79)(14 78)(15 77)(16 76)(17 75)(18 74)(19 73)(20 72)(21 71)(22 70)(23 69)(24 68)(25 67)(26 66)(27 65)(28 64)(29 108)(30 107)(31 106)(32 105)(33 104)(34 103)(35 102)(36 101)(37 100)(38 99)(39 98)(40 97)(41 96)(42 95)(43 94)(44 93)(45 92)(46 91)(47 90)(48 89)(49 88)(50 87)(51 86)(52 85)(53 112)(54 111)(55 110)(56 109)(113 141)(114 168)(115 167)(116 166)(117 165)(118 164)(119 163)(120 162)(121 161)(122 160)(123 159)(124 158)(125 157)(126 156)(127 155)(128 154)(129 153)(130 152)(131 151)(132 150)(133 149)(134 148)(135 147)(136 146)(137 145)(138 144)(139 143)(140 142)(169 209)(170 208)(171 207)(172 206)(173 205)(174 204)(175 203)(176 202)(177 201)(178 200)(179 199)(180 198)(181 197)(182 224)(183 223)(184 222)(185 221)(186 220)(187 219)(188 218)(189 217)(190 216)(191 215)(192 214)(193 213)(194 212)(195 211)(196 210)

G:=sub<Sym(224)| (1,161,64,122)(2,123,65,162)(3,163,66,124)(4,125,67,164)(5,165,68,126)(6,127,69,166)(7,167,70,128)(8,129,71,168)(9,141,72,130)(10,131,73,142)(11,143,74,132)(12,133,75,144)(13,145,76,134)(14,135,77,146)(15,147,78,136)(16,137,79,148)(17,149,80,138)(18,139,81,150)(19,151,82,140)(20,113,83,152)(21,153,84,114)(22,115,57,154)(23,155,58,116)(24,117,59,156)(25,157,60,118)(26,119,61,158)(27,159,62,120)(28,121,63,160)(29,190,95,203)(30,204,96,191)(31,192,97,205)(32,206,98,193)(33,194,99,207)(34,208,100,195)(35,196,101,209)(36,210,102,169)(37,170,103,211)(38,212,104,171)(39,172,105,213)(40,214,106,173)(41,174,107,215)(42,216,108,175)(43,176,109,217)(44,218,110,177)(45,178,111,219)(46,220,112,179)(47,180,85,221)(48,222,86,181)(49,182,87,223)(50,224,88,183)(51,184,89,197)(52,198,90,185)(53,186,91,199)(54,200,92,187)(55,188,93,201)(56,202,94,189), (1,169,64,210)(2,103,65,37)(3,171,66,212)(4,105,67,39)(5,173,68,214)(6,107,69,41)(7,175,70,216)(8,109,71,43)(9,177,72,218)(10,111,73,45)(11,179,74,220)(12,85,75,47)(13,181,76,222)(14,87,77,49)(15,183,78,224)(16,89,79,51)(17,185,80,198)(18,91,81,53)(19,187,82,200)(20,93,83,55)(21,189,84,202)(22,95,57,29)(23,191,58,204)(24,97,59,31)(25,193,60,206)(26,99,61,33)(27,195,62,208)(28,101,63,35)(30,155,96,116)(32,157,98,118)(34,159,100,120)(36,161,102,122)(38,163,104,124)(40,165,106,126)(42,167,108,128)(44,141,110,130)(46,143,112,132)(48,145,86,134)(50,147,88,136)(52,149,90,138)(54,151,92,140)(56,153,94,114)(113,188,152,201)(115,190,154,203)(117,192,156,205)(119,194,158,207)(121,196,160,209)(123,170,162,211)(125,172,164,213)(127,174,166,215)(129,176,168,217)(131,178,142,219)(133,180,144,221)(135,182,146,223)(137,184,148,197)(139,186,150,199), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28)(29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112)(113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140)(141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168)(169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196)(197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224), (1,63)(2,62)(3,61)(4,60)(5,59)(6,58)(7,57)(8,84)(9,83)(10,82)(11,81)(12,80)(13,79)(14,78)(15,77)(16,76)(17,75)(18,74)(19,73)(20,72)(21,71)(22,70)(23,69)(24,68)(25,67)(26,66)(27,65)(28,64)(29,108)(30,107)(31,106)(32,105)(33,104)(34,103)(35,102)(36,101)(37,100)(38,99)(39,98)(40,97)(41,96)(42,95)(43,94)(44,93)(45,92)(46,91)(47,90)(48,89)(49,88)(50,87)(51,86)(52,85)(53,112)(54,111)(55,110)(56,109)(113,141)(114,168)(115,167)(116,166)(117,165)(118,164)(119,163)(120,162)(121,161)(122,160)(123,159)(124,158)(125,157)(126,156)(127,155)(128,154)(129,153)(130,152)(131,151)(132,150)(133,149)(134,148)(135,147)(136,146)(137,145)(138,144)(139,143)(140,142)(169,209)(170,208)(171,207)(172,206)(173,205)(174,204)(175,203)(176,202)(177,201)(178,200)(179,199)(180,198)(181,197)(182,224)(183,223)(184,222)(185,221)(186,220)(187,219)(188,218)(189,217)(190,216)(191,215)(192,214)(193,213)(194,212)(195,211)(196,210)>;

G:=Group( (1,161,64,122)(2,123,65,162)(3,163,66,124)(4,125,67,164)(5,165,68,126)(6,127,69,166)(7,167,70,128)(8,129,71,168)(9,141,72,130)(10,131,73,142)(11,143,74,132)(12,133,75,144)(13,145,76,134)(14,135,77,146)(15,147,78,136)(16,137,79,148)(17,149,80,138)(18,139,81,150)(19,151,82,140)(20,113,83,152)(21,153,84,114)(22,115,57,154)(23,155,58,116)(24,117,59,156)(25,157,60,118)(26,119,61,158)(27,159,62,120)(28,121,63,160)(29,190,95,203)(30,204,96,191)(31,192,97,205)(32,206,98,193)(33,194,99,207)(34,208,100,195)(35,196,101,209)(36,210,102,169)(37,170,103,211)(38,212,104,171)(39,172,105,213)(40,214,106,173)(41,174,107,215)(42,216,108,175)(43,176,109,217)(44,218,110,177)(45,178,111,219)(46,220,112,179)(47,180,85,221)(48,222,86,181)(49,182,87,223)(50,224,88,183)(51,184,89,197)(52,198,90,185)(53,186,91,199)(54,200,92,187)(55,188,93,201)(56,202,94,189), (1,169,64,210)(2,103,65,37)(3,171,66,212)(4,105,67,39)(5,173,68,214)(6,107,69,41)(7,175,70,216)(8,109,71,43)(9,177,72,218)(10,111,73,45)(11,179,74,220)(12,85,75,47)(13,181,76,222)(14,87,77,49)(15,183,78,224)(16,89,79,51)(17,185,80,198)(18,91,81,53)(19,187,82,200)(20,93,83,55)(21,189,84,202)(22,95,57,29)(23,191,58,204)(24,97,59,31)(25,193,60,206)(26,99,61,33)(27,195,62,208)(28,101,63,35)(30,155,96,116)(32,157,98,118)(34,159,100,120)(36,161,102,122)(38,163,104,124)(40,165,106,126)(42,167,108,128)(44,141,110,130)(46,143,112,132)(48,145,86,134)(50,147,88,136)(52,149,90,138)(54,151,92,140)(56,153,94,114)(113,188,152,201)(115,190,154,203)(117,192,156,205)(119,194,158,207)(121,196,160,209)(123,170,162,211)(125,172,164,213)(127,174,166,215)(129,176,168,217)(131,178,142,219)(133,180,144,221)(135,182,146,223)(137,184,148,197)(139,186,150,199), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28)(29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112)(113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140)(141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168)(169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196)(197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224), (1,63)(2,62)(3,61)(4,60)(5,59)(6,58)(7,57)(8,84)(9,83)(10,82)(11,81)(12,80)(13,79)(14,78)(15,77)(16,76)(17,75)(18,74)(19,73)(20,72)(21,71)(22,70)(23,69)(24,68)(25,67)(26,66)(27,65)(28,64)(29,108)(30,107)(31,106)(32,105)(33,104)(34,103)(35,102)(36,101)(37,100)(38,99)(39,98)(40,97)(41,96)(42,95)(43,94)(44,93)(45,92)(46,91)(47,90)(48,89)(49,88)(50,87)(51,86)(52,85)(53,112)(54,111)(55,110)(56,109)(113,141)(114,168)(115,167)(116,166)(117,165)(118,164)(119,163)(120,162)(121,161)(122,160)(123,159)(124,158)(125,157)(126,156)(127,155)(128,154)(129,153)(130,152)(131,151)(132,150)(133,149)(134,148)(135,147)(136,146)(137,145)(138,144)(139,143)(140,142)(169,209)(170,208)(171,207)(172,206)(173,205)(174,204)(175,203)(176,202)(177,201)(178,200)(179,199)(180,198)(181,197)(182,224)(183,223)(184,222)(185,221)(186,220)(187,219)(188,218)(189,217)(190,216)(191,215)(192,214)(193,213)(194,212)(195,211)(196,210) );

G=PermutationGroup([[(1,161,64,122),(2,123,65,162),(3,163,66,124),(4,125,67,164),(5,165,68,126),(6,127,69,166),(7,167,70,128),(8,129,71,168),(9,141,72,130),(10,131,73,142),(11,143,74,132),(12,133,75,144),(13,145,76,134),(14,135,77,146),(15,147,78,136),(16,137,79,148),(17,149,80,138),(18,139,81,150),(19,151,82,140),(20,113,83,152),(21,153,84,114),(22,115,57,154),(23,155,58,116),(24,117,59,156),(25,157,60,118),(26,119,61,158),(27,159,62,120),(28,121,63,160),(29,190,95,203),(30,204,96,191),(31,192,97,205),(32,206,98,193),(33,194,99,207),(34,208,100,195),(35,196,101,209),(36,210,102,169),(37,170,103,211),(38,212,104,171),(39,172,105,213),(40,214,106,173),(41,174,107,215),(42,216,108,175),(43,176,109,217),(44,218,110,177),(45,178,111,219),(46,220,112,179),(47,180,85,221),(48,222,86,181),(49,182,87,223),(50,224,88,183),(51,184,89,197),(52,198,90,185),(53,186,91,199),(54,200,92,187),(55,188,93,201),(56,202,94,189)], [(1,169,64,210),(2,103,65,37),(3,171,66,212),(4,105,67,39),(5,173,68,214),(6,107,69,41),(7,175,70,216),(8,109,71,43),(9,177,72,218),(10,111,73,45),(11,179,74,220),(12,85,75,47),(13,181,76,222),(14,87,77,49),(15,183,78,224),(16,89,79,51),(17,185,80,198),(18,91,81,53),(19,187,82,200),(20,93,83,55),(21,189,84,202),(22,95,57,29),(23,191,58,204),(24,97,59,31),(25,193,60,206),(26,99,61,33),(27,195,62,208),(28,101,63,35),(30,155,96,116),(32,157,98,118),(34,159,100,120),(36,161,102,122),(38,163,104,124),(40,165,106,126),(42,167,108,128),(44,141,110,130),(46,143,112,132),(48,145,86,134),(50,147,88,136),(52,149,90,138),(54,151,92,140),(56,153,94,114),(113,188,152,201),(115,190,154,203),(117,192,156,205),(119,194,158,207),(121,196,160,209),(123,170,162,211),(125,172,164,213),(127,174,166,215),(129,176,168,217),(131,178,142,219),(133,180,144,221),(135,182,146,223),(137,184,148,197),(139,186,150,199)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28),(29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84),(85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112),(113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140),(141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168),(169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196),(197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224)], [(1,63),(2,62),(3,61),(4,60),(5,59),(6,58),(7,57),(8,84),(9,83),(10,82),(11,81),(12,80),(13,79),(14,78),(15,77),(16,76),(17,75),(18,74),(19,73),(20,72),(21,71),(22,70),(23,69),(24,68),(25,67),(26,66),(27,65),(28,64),(29,108),(30,107),(31,106),(32,105),(33,104),(34,103),(35,102),(36,101),(37,100),(38,99),(39,98),(40,97),(41,96),(42,95),(43,94),(44,93),(45,92),(46,91),(47,90),(48,89),(49,88),(50,87),(51,86),(52,85),(53,112),(54,111),(55,110),(56,109),(113,141),(114,168),(115,167),(116,166),(117,165),(118,164),(119,163),(120,162),(121,161),(122,160),(123,159),(124,158),(125,157),(126,156),(127,155),(128,154),(129,153),(130,152),(131,151),(132,150),(133,149),(134,148),(135,147),(136,146),(137,145),(138,144),(139,143),(140,142),(169,209),(170,208),(171,207),(172,206),(173,205),(174,204),(175,203),(176,202),(177,201),(178,200),(179,199),(180,198),(181,197),(182,224),(183,223),(184,222),(185,221),(186,220),(187,219),(188,218),(189,217),(190,216),(191,215),(192,214),(193,213),(194,212),(195,211),(196,210)]])

61 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	4A	4B	4C	4D	4E	4F	4G	4H	7A	7B	7C	8A	8B	8C	8D	14A	···	14I	28A	···	28F	28G	···	28R	56A	···	56L
order	1	2	2	2	2	2	2	4	4	4	4	4	4	4	4	7	7	7	8	8	8	8	14	···	14	28	···	28	28	···	28	56	···	56
size	1	1	1	1	14	14	56	2	2	4	4	8	28	28	28	2	2	2	4	4	28	28	2	···	2	4	···	4	8	···	8	4	···	4

61 irreducible representations

dim	1	1	1	1	1	1	1	1	2	2	2	2	2	2	2	2	2	2	4	4	4	4	4
type	+	+	+	+	+	+	+	+	+	+	+	+	+		+	+	+	+	-	+	+
image	C₁	C₂	C₂	C₂	C₂	C₂	C₂	C₂	D₄	D₄	D₄	D₄	D₇	SD₁₆	D₁₄	D₁₄	D₁₄	D₂₈	C₈.C₂²	D₄×D₇	D₄×D₇	D₇×SD₁₆	Q₁₆⋊D₇
kernel	Q₈⋊₂D₂₈	C₁₄.Q₁₆	D₁₄⋊C₈	C₇×Q₈⋊C₄	C₄⋊D₂₈	C₂×C₅₆⋊C₂	C₂×Q₈⋊D₇	C₂×Q₈×D₇	Dic₁₄	C₂×Dic₇	C₇×Q₈	C₂²×D₇	Q₈⋊C₄	D₁₄	C₄⋊C₄	C₂×C₈	C₂×Q₈	Q₈	C₁₄	C₄	C₂²	C₂	C₂
# reps	1	1	1	1	1	1	1	1	2	1	2	1	3	4	3	3	3	12	1	3	3	6	6

Matrix representation of Q₈⋊₂D₂₈ ►in GL₆(𝔽₁₁₃)

1	0	0	0	0	0
0	1	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	1	106
0	0	0	0	81	112

112	0	0	0	0	0
0	112	0	0	0	0
0	0	112	0	0	0
0	0	0	112	0	0
0	0	0	0	20	111
0	0	0	0	31	93

76	106	0	0	0	0
2	37	0	0	0	0
0	0	103	24	0	0
0	0	89	1	0	0
0	0	0	0	109	101
0	0	0	0	86	4

76	106	0	0	0	0
34	37	0	0	0	0
0	0	103	24	0	0
0	0	10	10	0	0
0	0	0	0	4	12
0	0	0	0	27	109

G:=sub<GL(6,GF(113))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,81,0,0,0,0,106,112],[112,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,20,31,0,0,0,0,111,93],[76,2,0,0,0,0,106,37,0,0,0,0,0,0,103,89,0,0,0,0,24,1,0,0,0,0,0,0,109,86,0,0,0,0,101,4],[76,34,0,0,0,0,106,37,0,0,0,0,0,0,103,10,0,0,0,0,24,10,0,0,0,0,0,0,4,27,0,0,0,0,12,109] >;

Q₈⋊₂D₂₈ in GAP, Magma, Sage, TeX

Q_8\rtimes_2D_{28}

% in TeX

G:=Group("Q8:2D28");

// GroupNames label

G:=SmallGroup(448,340);

// by ID

G=gap.SmallGroup(448,340);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,232,254,219,58,851,438,102,18822]);

// Polycyclic

G:=Group<a,b,c,d|a^4=c^28=d^2=1,b^2=a^2,b*a*b^-1=c*a*c^-1=d*a*d=a^-1,c*b*c^-1=d*b*d=a*b,d*c*d=c^-1>;

// generators/relations

G = Q8⋊2D28 order 448 = 26·7

1st semidirect product of Q8 and D28 acting via D28/D14=C2

G = Q₈⋊₂D₂₈ order 448 = 2⁶·7

1^st semidirect product of Q₈ and D₂₈ acting via D₂₈/D₁₄=C₂